Question

In a study of treatments for very painful "cluster" headaches, 159 patients were treated with oxygen and 147 other patients were given a placebo consisting of ordinary air. Among the 159 patients in the oxygen treatment group, 124 were free from headaches 15 minutes after treatment. Among the 147 patients given the placebo, 21 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Complete parts (a) through (c) below. a. Test the claim using a hypothesis test. Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hypothesis test? Ho P1 P2 H P1P2 O B. Ho: P1= P2 H1 P1P2 OE. Ho: P12P2 H P1P2 OA. Ho P1 P2 H p1 P2 O F. Ho: P1 P2 D. Ho: P1 SP2 H P1 P2 H: P1P2

(a) Identify Test Statistic--------------------? (Round to two decimal places as needed)

(b) Identify the P Value------------------------?

(c) Identify Critical Value-----------------------?

(d) Construct appropriate Confidence interval--------------------?

(e) State Conclusion-----------------------?

Answer 1

a) $\widehat p1$ = 124/159 = 0.78

  $\widehat p2$ = 21/147 = 0.143

The pooled sample proportion (P) = ($\widehat p1$ * n1 + $\widehat p2$ * n2)/(n1 + n2) = (0.78 * 159 + 0.143 * 147)/(159 + 147) = 0.474

SE = sqrt(P(1 - P)(1/n1 + 1/n2))

= sqrt(0.474 * (1 - 0.474) * (1/159 + 1/147))

= 0.057

The test statistic z = ($\widehat p1 - \widehat p2$)/SE

= (0.78 - 0.143)/0.057 = 11.18

b) P-value = P(Z > 11.18)

= 1 - P(Z < 11.18)  

= 1 - 1 = 0

c) At alpha = 0.05, the critical value is z_0.95 = 1.645

d) The 95% confidence interval is.

($\widehat p1 - \widehat p2$) +/- z_0.025 * SE

= (0.78 - 0.143) +/- 1.96 * 0.057

= 0.637 +/- 0.1117

= 0.5253, 0.7487

E) Since the P-value is less than the significance level alpha = 0.05, so we should reject the null hypothesis.